reserve c, c1, c2, d, d1, d2, e, y for Real,
  k, n, m, N, n1, N0, N1, N2, N3, M for Element of NAT,
  x for set;

theorem
  ex s, s1 being eventually-positive Real_Sequence st s = seq_a^(2,1,0)
  & s1 = seq_a^(2,1,1) & Big_Oh(s) = Big_Oh(s1)
proof
  set g = seq_a^(2,1,1);
  set f = seq_a^(2,1,0);
  set h = f/"g;
  reconsider f as eventually-positive Real_Sequence;
  reconsider g as eventually-positive Real_Sequence;
  take f,g;
  thus f = seq_a^(2,1,0) & g = seq_a^(2,1,1);
A1: now
    let n;
A2: g.n = 2 to_power (1*n + 1) by Def1;
    f.n = 2 to_power (1*n + 0) by Def1;
    then h.n = (2 to_power n)/g.n by Lm4
      .= 2 to_power (n-(n+1)) by A2,POWER:29
      .= 2 to_power (0+-1)
      .= 1 / 2 to_power 1 by POWER:28
      .= 1 / 2 by POWER:25
      .= 2";
    hence h.n = 2";
  end;
A3: now
    let p be Real such that
A4: p > 0;
     reconsider N = 0 as Nat;
    take N;
    let n be Nat;
A5:  n in NAT by ORDINAL1:def 12;
    assume n >= N;
    |.h.n-2".| = |.2"-2".| by A1,A5
      .= 0 by ABSVALUE:2;
    hence |.h.n-2".| < p by A4;
  end;
  then
A6: h is convergent by SEQ_2:def 6;
  then lim h > 0 by A3,SEQ_2:def 7;
  hence thesis by A6,ASYMPT_0:15;
end;
